decimal equivalents of fractions -- theory

Question

In a ratio of integers, the corresponding decimal value either terminates or repeats in no more digits than the size of the denominator.

 

Could you please explain in detail why this is true?

 

Thank you so much in advance! I'm struggling to explain this to my daughter (we are going through a Connected Mathematics chapter together). 

 

Elena

Answer 1

Because not all of them will divide evenly.

 

To change a fraction into a decimal number, you divide the top number 

by the bottom number. That is, the top number goes into the division box.

You then put the decimal point after the top number and add as many

zeros on the back end as you like. Upon division, the calculations

will either stop with a remainder of zero or a repeating pattern will

occur.                                                              

                                                                   0.75

 

                                                                  _______

For example:   3/4 =   3 divided by 4  =     4 | 3.00000

                                                                     28

                                                                     --------

                                                                       20

                                                                       20

                                                                       ----

                                                                         0

 

On the other hand, the fraction 1/3 = 0.33333333.... = 0.3

 

where the bar indicates that the digits repeat

 

23/99 = 0.2323232323 = 0.23

 

The 7s are interesting creatures. They exhibit the property you are speaking of.

There are no more repeating digits than the denominator.

1/7 = 0.142857 <--- 6 repeating digits

 

This is because the remainder must be LESS than the number you are dividing by.